A honeycomb is said to have regular tessellations. Looking at this image, can you identify what it means to tessellate?

In this concept, you will learn to identify tessellations.

Tessellations

You can use translations and reflections to make patterns with geometric figures called tessellations. A tessellation is a pattern in which geometric figures repeat without any gaps between them. In other words, the repeated figures fit perfectly together. They form a pattern that can stretch in every direction on the coordinate plane.

Take a look at the tessellations below. The tessellation can go on and on forever.

You can create tessellations by moving a single geometric figure. You can perform transformations such as translations and rotations to move the figure so that the original and the new figure fit together.

How do you know that a figure will tessellate?

If the figure is the same on all sides, it will fit together when it is repeated. Figures that tessellate tend to be regular polygons. Regular polygons have congruent straight sides. When you rotate or slide a regular polygon, the side of the original figure and the side of its translation will match. Not all geometric figures can tessellate. When you translate or rotate them, their sides do not fit together.

Remember this rule and you will know whether a figure will tessellate or not! Think about whether or not there will be gaps in the pattern as you move a figure.

First, trace the figure on a piece of stiff paper and then cut it out. This will let you perform translations easily so you can see how best to repeat the figure to make a tessellation.

This figure is exactly the same on all sides, so you do not need to rotate it to make the pieces fit together. Instead, let’s try translating it.

Next, trace the figure. Then slide the cutout so that one edge of it lines up perfectly with one edge of the figure you drew. Trace the cutout again.

Then, line the cutout up with another side of the original figure and trace it. As you add figures to the pattern, the hexagons will start making themselves!

Check to make sure that there are no gaps in your pattern. All of the edges should fit perfectly together. You should be able to continue sliding and tracing the hexagon forever in all directions. You have made a tessellation!

Examples

Example 1

Earlier, you were given a problem about tessellating honeycomb.

To tessellate means that congruent figures are put together to create a pattern where there aren’t any gaps or spaces in the pattern. Figures can be put side by side and/or upside down to create the pattern. The pattern is called a tessellation. If you look at the honeycomb, you can see that hexagons are tessellating.

Regular polygons will tessellate as long as one of their interior angles is a factor of 360°. To find the interior angles of a regular polygon, you use the formula:

\begin{align*}\frac{180(n-2)}{n}\end{align*}

where \begin{align*}n\end{align*} is the number of sides in the polygon.

One interior angle of a regular pentagon (5 sides) is \begin{align*}\frac{180(5-2)}{5} = \frac{540}{5} - 108^\circ\end{align*}. 360° is not divisible by 108°. \begin{align*}\frac{360}{108} = 3.333\end{align*}. Because 108 is not a factor of 360, a regular pentagon will not tessellate.

A regular hexagon (like in the honeycomb) does tessellate. One interior angle of a regular hexagon is \begin{align*}\frac{180(6-2)}{6} = \frac{720}{6} = 120^\circ\end{align*}. 360° is divisible by 120°. \begin{align*}\frac{360}{120} = 3\end{align*}. Because 120 is a factor of 360, a regular hexagon will tessellate.

Example 2

Draw a tessellation of equilateral triangles.

In an equilateral triangle each angle is 60°. Therefore, six triangles will perfectly fit around each point to make a hexagon.